Quadrilateral Meshes with Bounded Minimum Angle

Transcription

1 Quadrilateral Meshes with Bounded Minimum Angle F. Betul Atalay 1, Suneeta Ramaswami 2,, and Dianna Xu 3 1 St. Joseph s Uniersity, Philadelphia, PA. fatalay@sju.edu 2 Rutgers Uniersity, Camden, NJ. rsuneeta@camden.rutgers.edu 3 Bryn Mawr College, Bryn Mawr, PA. dxu@cs.brynmawr.edu Summary. This paper presents an algorithm that utilizes a quadtree to construct a strictly conex quadrilateral mesh for a simple polygonal region in which no newly created angle is smaller than (= arctan( 1 )). This is the first known result, to the best of our knowledge, on quadrilateral mesh generation with a proable guarantee on the minimum 3 angle. 1 Introduction The generation of quadrilateral meshes with proable guarantees on mesh quality poses seeral interesting open questions. While theoretical properties of triangle meshes are well understood [4, 8, 9, 7, 11, 12, 15, 14], much less is known about algorithms for proably good quadrilateral meshes. Analysts, howeer, prefer quadrilateral and hexahedral meshes for better solution quality in numerous applications [1, 2, 6, 10, 16]. This is because they hae better conergence properties, and hence lower approximation errors, in finite element methods for solutions to systems of partial differential equations. Quadrilateral meshes also offer lower mesh complexity, and better directionality control for anisotropic meshing. For stable analytical results, howeer, it is critical to construct meshes with certain quality guarantees. Specifically, algorithms that construct well-shaped elements by proiding bounds on minimum and maximum angles hae much practical alue. Techniques such as paing [5] work well in practice, but do not gie proable angle guarantees. Circlepacking techniques hae been used to construct quadrangulations with no angles larger than 120 for polygon interiors [3], but with no bound on smallest angle. An algorithm to construct linear-sized strictly conex quadrilateral meshes for arbitrary planar straight line graphs is gien in [13]. Our contribution. In this paper, we present a new algorithm to generate quadrilateral meshes for simple polygonal regions, possibly with holes, with a proable guarantee on the minimum angle. We use quadtrees to show that no newly created angle in the quadrilateral mesh is smaller than The quadrilaterals are strictly Research partially supported by NSF Grant

2 2 F. Betul Atalay, Suneeta Ramaswami, and Dianna Xu conex, i.e., the maximum angle is strictly less than 180. This is the first known quadrilateral mesh generation algorithm with a proable bound on the minimum angle. (Quadtrees hae been used to gie triangular meshes without small angles for point sets and polygons in 2D [4], and octrees hae been utilized to construct tetrahedral meshes with bounded aspect-ratio elements for polyhedra [12].) In Section 2, we use quadtrees to construct a quadrilateral mesh for a point set in which the minimum angle is bounded below by 45 arctan( 1 3 ) = We then describe in Section 3 an algorithm that adapts the guaranteed-quality mesh of polygon ertices to polygon edges to construct a quadrilateral mesh for the interior of a simple polygon (possibly with holes) in which new angles (angles other than those determined by the input) are bounded below by arctan( 1 3 ) = Throughout this paper, we use the shorter terms quadrangulate and quadrangulation instead of quadrilateralize and quadrilateralization. We also sometimes use the word quad for quadrilateral. Steiner points are additional points, other than those proided by the input, inserted during the mesh generation process. 2 Point Set Mesh with Bounded Minimum Angle We first describe an algorithm to construct a quadrilateral mesh with a minimum angle bound of for a gien point set X. 2.1 Construction of the quadtree Gien a point set X, we construct a quadtree for X with the following separation and balancing conditions. These conditions are similar to those in [4], but adapted to particular requirements for quadrilateral (rather than triangle) meshing. A. Split a cell C (with side length l) containing at least one point if it is crowded. A cell is crowded if one or more of the following conditions hold: 1. it contains more than one point from X. 2. one of the extended neighbors is split (an extended neighbor is a cell of same size sharing either a side or corner of C). 3. it contains a point with a nearest neighbor less than 2 2l units away. B. When a crowded cell C is split, split those extended neighbors of C that share an edge or corner with a child of C containing an original point in X. C. The final quadtree is balanced so that the edge lengths of two adjacent cells differ at most by a factor of 2 (neighbors of C with side length l hae length l/2 or 2l). Obsere that in a quadtree with the aboe separation and balancing conditions, a cell containing a point from X is guaranteed to be surrounded by 8 empty cells of the same size. We refine the quadtree decomposition further to do the following: Split each of these eight empty quadtree cells into 2 2 cells and rebalance the quadtree. This conerts the original 3 3 grid around eery point p X into a 6 6 grid. Furthermore, now p lies at the center of a 5 5 equal-sized grid (outlined in bold in figure), and is surrounded by twenty-four empty quadtree cells of the same size. There are two reasons for this refinement step:

3 Quadrilateral Meshes with Bounded Minimum Angle 3 1. The final step of our algorithm to construct a quadrilateral mesh for X consists of warping a Steiner point in the mesh to an original point p X (Section 2.4). This step is simplified considerably due to the refinement. 2. The algorithm to construct a quadrilateral mesh for non-acute polygons (Section 3.1) uses the 5 5 grid to mesh the region around polygon ertices. We construct a quadrilateral mesh with bounded minimum angle for X by placing Steiner points in the interior of the quadtree cells. The placement of the Steiner points is determined by identifying and applying templates to the quadtree decomposition. A leaf of the quadtree is an unsplit cell and we refer to these as 1-cells in our discussion. A template is applied to each internal node of the quadtree. PSfrag replacements 2.2 The templates A template is labeled by the number of children of a quadtree node that are 1-cells. Hence we hae 6 template configurations, for nodes with zero (T (0) ), one (T (1) ), two, three (T (3) ) or four (T (4) ) 1-cell children. Nodes with two 1-cell children hae two layouts, T (2a) and T (2b). Templates at the deepest leel of subdiision. The templates at the deepest leel of subdiision are shown in Fig. 1. Note that, all other possible configurations are symmetric to the depicted ones. In order to quadrangulate a template, first, a Steiner point is placed at the center of each quadtree cell. These points are denoted with full circles. We then place extra Steiner points, which are denoted by empty circles in the figure, for one of two reasons: (i) In T (1), the top-left extra point and in T (2b) the middle extra point are added to be able to quadrangulate properly within the template. (ii) The remaining extra points are added in the 1-cells, halfway on the diagonal between the center Steiner point and the outer cell corner. The reason for adding the second type of Steiner points is that after an internal node is quadrangulated, it will proide a polygonal chain with an een number of points (we will call them een-connector chains) to which its neighbors can connect. T (0) T (1) T (2a) T (2b) T (3) T (4) Fig. 1: Templates at the deepest leel of the subdiision. General Templates. Our recursie algorithm applies templates to all internal nodes starting with the deepest ones. We generalize the templates to apply to an arbitrarily deep internal node as shown in Fig. 2. In general, when a template is applied to an internal node, its children which are not 1-cells hae already had templates applied to them. Each such child has been quadrangulated internally and proides eenconnector chains on all four sides. The corresponding endpoints of two neighboring

4 4 F. Betul Atalay, Suneeta Ramaswami, and Dianna Xu chains are then connected to construct a polygon with guaranteed een number of ertices which can therefore be quadrangulated. We name this process stitching, illustrated by the cross-hatched regions in Fig. 2. The processed internal nodes are depicted as blackboxes with een-connector chains at each side. Note that the placement of a chain s endpoint does not necessarily correspond to the exact location of the endpoint within the actual cell, due to the existence of type (ii) Steiner points. PSfrag replacements (a) T (0) (f) T (4) (b) T (1) (c) T (2a) (d) T (2b) (e) T (3) een connector 2 connector quadrangulated cell stitched region deleted ertex Fig. 2: General templates at arbitrary leel of subdiision Labeling the chains. Children quadrants of a cell are labeled C 0, C 1, C 2, and C 3 in counterclockwise order starting from the northeast quadrant. The four chains surrounding a processed quadrant C i are labeled l i, r i, t i and b i (see figure). t i C 1 C 0 l i C i C 2 C 3 b i r i 2.3 The algorithm The recursie procedure applytemplate that applies a template to an internal node is presented in the code block gien in Fig. 3. It is initially called with the root node of the quadtree. Note that the algorithm is presented only with respect to the depicted configurations of the templates. Symmetric configurations are handled similarly. Stitching Chains. Procedure stitchchains connects the four endpoints of two neighboring een-connector chains and quadrangulates the resulting polygon. Note that such a polygon is guaranteed to hae an een number of ertices on the boundary. The algorithm is illustrated in Fig 4. Procedure stitchchains is only called if the current template is of type T (0), T (1) or T (2a). The action of this procedure is also illustrated by the crosshatched areas in Fig. 2(a), (b) and (c). The quadrangulation process diides the chains into half chains, each of which spans the corresponding edge of a child quadrant. These half chains are then recursiely stitched. Although the een-connector chains can be arbitrarily long, at the base case there are only four types of chains: chains with 2, 4, 6 or 8 connectors.

5 Quadrilateral Meshes with Bounded Minimum Angle 5 applytemplate(quadtreenode N) templatetype whichtemplate(n) for C i children(n) if C i is not a 1-cell (l i, r i, t i, b i) applytemplate(c i) else construct (l i, r i, t i, b i) for C i. switch (templatetype) case T (0) : stitchchains(l 0, r 1), stitchchains(b 1, t 2) stitchchains(r 2, l 3), stitchchains(t 3, b 0) case T (1) : stitchchains(l 0, r 1), stitchchains(b 1, t 2) Place Steiner points and quadrangulate per Fig. 2(b). case T (2a) : stitchchains(r 2, l 3) Place Steiner points and quadrangulate per Fig. 2(c). case T (2b) : Place Steiner points and quadrangulate per Fig. 2(d). case T (3) : Place Steiner points and quadrangulate per Fig. 2(e). case T (4) : Place Steiner points and quadrangulate per Fig. 2(f). return (l 1 + l 2, r 0 + r 3, t 1 + t 0, b 2 + b 3) Fig. 3: applytemplate quadrangulates node N. stitchchains(chain ch 1, Chain ch 2) switch (length(ch 1), length(ch 2)) case (2 2), (2 4), (4 2): Apply appropriate base case from Fig. 5. case (2 6), (2 8)): Apply appropriate base case from Fig. 6. default: (f 1, s 1) gethalfchains(ch 1) (f 2, s 2) gethalfchains(ch 2) stitchchains(f 1, f 2) stitchchains(s 1, s 2) Fig. 4: stitchchains stitches two eenconnector chains, one from each of the two neighbor cells sharing an edge. Figs. 5-6 illustrate how the base-case chains are stitched (the stitching edges are dotted). Symmetric cases are not listed in the illustrations. (1) (2) (3) (4) (5) (6) Fig. 5: Stitching 2-2 and 2-4 or 4-2 connector chains. (7) (1) (2) Fig. 6: Stitching 2-6 and 2-8 connector chains. (3) 2.4 Angle Bounds Minimum Angle. We analyze the minimum angle bound in each of applytemplate, the base case of stitchchains, and the recursie step of stitchchains. General templates: By construction, the minimum angle appears in templates T (1) and T (2b) and equals 45 arctan( 1 3 ) = (illustrated in Fig. 7). Stitching base case: The base cases of stitching generate the same minimum angle of 45 arctan( 1 3 ) = which can be found in in Fig. 5-(5).

6 6 F. Betul Atalay, Suneeta Ramaswami, and Dianna Xu Stitching merging step: After the corresponding half-chains are stitched in the recursie step of stitchchains, a middle quad is formed by the four end points of the stitched half-chains. This middle quad gies a minimum angle of 2 arctan( 1 4 ) = See Fig. 7. Recall that these four points are by construction on the two diagonals that cross at the center of four quadtree quadrants. Furthermore, they are either 45 arctan( 1 3 ) = arctan( 1 3 ) = arctan( 1 4 ) = Fig. 7: Minimum angle bounds at the center of the quadtree quadrant, or halfway down the diagonal from the center. The worst-case configuration is illustrated in Fig. 7. This results from connecting any Fig. 5-(5) connector chain with an inerted ersion of itself. Degenerate quads. In the stitching cases illustrated by Fig. 5-(7), Fig. 6-(1), Fig. 6- (2) as well as template T (2a) (Fig. 1), there are degenerate quads with two edges on a straight line. In all cases, the 180 ertex is connected to a third ertex on the other side of the degenerate quad, by construction. This allows perturbation of the degenerate ertex along the third edge, which reduces the 180 angle and increases the other two, thus eliminating the degenerate quad. Maximum Angle. The quadrilaterals generated by our algorithms (including the one in Section 3) are strictly conex; i.e., the maximum angle is bounded away from 180. The perturbation used to handle degenerate quads (aboe) implies the maximum angle is less than (180 ɛ) for some ɛ > 0. We conjecture the alue of ɛ can be bounded from below, but do not explicitly address that question in this paper. p Warp Warping to Original Points. After the construction of the quadrilateral mesh using quadtree cell centers and extra points as Steiner points, we warp certain mesh ertices to the original points from the input point set X. Recall that the quadtree splitting rules of Section 2.1 ensure that the quadtree cell containing an original point p X is surrounded by twenty-four empty quadtree cells of the same size. Moreoer, the eight empty cells immediately surrounding p do not contain any extra points. Therefore, the warping step simply consists of translating the Steiner point in p s cell to p, along with all the incident edges. The worst-case minimum angle arising from these nine cells after the warping step is = In summary, we hae shown the following result: Theorem 1. Gien a quadtree decomposition with N quadtree cells satisfying the point set separation conditions for a point set X, applytemplate constructs a mesh for X with at most 3N quadrilaterals in which eery angle is at least degrees. Obsere that the alue of N in the aboe theorem depends on the geometry of the point set as well as the size of the point set. Due to the point set separation conditions, which are deried from [4] and as was shown there, the size of the quadtree decomposition increases as the distance between the closest pair of points decreases.

7 Quadrilateral Meshes with Bounded Minimum Angle 7 3 Polygon Mesh with Bounded Minimum Angle Gien a simple polygon P, possibly with holes, with ertex set X, we gie an algorithm to construct a quadrilateral mesh for P and its interior in which no new angle is larger than The basidea behind the algorithm is to first construct a guaranteed-quality mesh for X as described in the preious section, and then adapt this mesh to incorporate the edges of P. We use δp to refer to the polygon boundary, and P to refer to the union of the boundary as well as interior. We describe in Section 3.1 a proably good algorithm to construct a quadrilateral mesh with bounded minimum angle for a simple polygon P in which all interior angles are non-acute (i.e., greater than or equal to 90 ). In Section 3.2, we describe how to handle acute angles. 3.1 Non-acute Simple Polygons Let P be a non-acute polygon with ertex set X and edges oriented counterclockwise about the boundary. Let QT be a quadtree decomposition of X satisfying the point set separation conditions of Section 2.1. Let Q be a quadrilateral mesh for X with minimum angle 26.57, as guaranteed by Theorem 1. In this section, we describe a method to adapt Q to δp to create a constrained quadrilateral mesh for P. In a constrained quadrilateral mesh, we allow Steiner points to be inserted on δp as well, so that the union of the finite elements of the mesh is equal to P. We start by describing an algorithm to adapt Q to include a single edge of P. In order to use this algorithm on all edges of P, QT must satisfy certain polygon edge separation conditions, which are discussed towards the end of the section. We conclude the section by describing how to construct the final constrained mesh for P by adapting to the regions around the ertices. Inserting an edge into Q Consider an edge e = (a, b) of P oriented from a to b, where a, b X. Assume that e makes an angle between 45 and 45 with the positie x axis (if not, orient the x axis so that this is the case). We say that a point lies aboe e if it lies in the open halfspace to the left of the oriented line through e. We use e to define two chains of edges from Q and QT, as described below: (i) e intersects quadrilaterals of Q. Edges of these quadrilaterals are used to define a chain of edges called the quadrangulation chain α associated with e. (ii) e intersects quadtree cells of QT. The centers of these cells are used to define a chain of edges called the quadtree chain β associated with e. Quadrangulation Chain. Let q 1, q 2,..., q k be the quadrilaterals of Q intersected by e in left to right order as traersed from a to b (since the quadrilaterals are conex, each q i is unique). Let E i be the edges of q i that lie entirely aboe e. E i may hae 0, 1, or 2 edges. If E i has two edges, they are listed in clockwise order about q i. Then the quadrangulation chain α is defined as α = E 1 E 2... E k, where represents edge concatenation. See Fig. 8 for an example of a quadrangulation chain, in

8 8 F. Betul Atalay, Suneeta Ramaswami, and Dianna Xu which E 1 has 1 edge, E 2 has 2 edges, and E 3 α q 9 q 11 q 8 b has 0 edges. Note that the same edge may repeat q 6 q 7 q 10 twice in α (the repetitions always appear consecutiely) and such an edge is incident to q q 4 q 2 q 5 i that a q q 3 1 has E i = 0. For example, the quadrangulation chain in Fig. 8 has three repeating edges, which Fig. 8: Quadrangulation chain α. are incident to the quadrilaterals q 3, q 5 and q 10. If we drop the repetitions from α, then α is a weakly simple polygonal chain. A ertex belongs to an edge if it is one of the endpoints of the edge. We say that α if is a ertex of Q and belongs to one of the edges of α. If we quadrangulate the region bounded by α and e by adding Steiner points either in the interior of the region or on e itself, the resulting quadrangulation is compatible with Q (since edges of α are edges in Q). Howeer, in order to quadrangulate the region with the desired angle bounds, we need to know more about the geometry of α. The quadtree chain, described below, allows us to establish the required geometric properties for α. Quadtree Chain. In the remainder of the paper, we use the same symbol to refer to a quadtree cell as well as its center wheneer the meaning is clear from the context. Gien a cell c, N(c), W (c), and E(c) denote, respectiely, the set of north, west, and east neighbor cells of c (note that each set has at most two elements in it because of the balancing conditions for QT ). Let C be the set of cell centers of quadtree cells in QT that are intersected by e. C does not include the starting or ending cells (i.e., the cells containing a and b, respectiely). Let θ be the angle (in degrees) that e makes with the positie x axis. The quadtree chain β is defined as follows: 1. If c C and c lies aboe e, then c belongs to β. 2. If c C and c lies below e, then N(c) β. Note that the centers of cells in N(c) must lie aboe e under our assumption that 45 θ If c C, c lies below e, and 0 θ 45 (resp., 45 θ < 0), then a cell center in W (c) (resp., E(c)) belongs to β if it lies aboe e. Let {c 1, c 2,..., c m } be the cell centers in β in lexicographically sorted (by x, then y) order. Recall that Q is constructed from the quadtree decomposition QT of X. The oerall approach to incorporating edge e into Q is summarized below: (A) We first show that quadtree chain β is a subset of quadrangulation chain α. (B) This fact allows us, in turn, to exploit the structure proided by QT and our algorithm from Section 2 to identify a small number of possible ways in which two consecutie points and +1 of β can be connected along the chain α. We use α i, 1 i m 1, to refer to the subchain of α starting at and ending at +1. α i may lie under +1. In this case, we choose instead a chain of edges in Q lying aboe +1 in order to simplify the final quadrangulation step in part (C) below. (C) Finally, we quadrangulate the region bounded by e and α by breaking it into smaller sub-regions defined by perpendicular projections from and +1 onto

9 Quadrilateral Meshes with Bounded Minimum Angle 9 e. The case analysis form part (B) is then used to proe a minimum angle guarantee of for the quadrangulation of each subregion. Lemmas 1-3 are required for steps (A)-(C) and stated here without proof. Lemma 1. For 1 i m, α. That is, eery cell center in the quadtree chain belongs to the quadrangulation chain. Lemma 2. For 1 i m 1, and +1 are edge or corner neighbors in QT. Lemma 3. Let i be the ertical projection of on e. For 1 i m, the segment i does not intersect α. Lemmas 1 and 3 imply that the edge sequence ( i, ) α i (+1, i+1 ) ( i+1, i ) defines a simple polygon for all 1 i m 1. Call this polygon A i. We now use Lemma 2 to proe that α i is composed of at most four edges. This is done ia a case analysis on the ways in which and +1 are connected in Q. Lemma 4. The number of edges in α i is at most four. Proof: We know from Lemma 2 that and +1 are either edge or corner neighbors in QT. We consider each case separately. Our case analysis only depicts α i with two or more edges (i.e., when and +1 are not directly connected). Let s i, 1 i m refer to the size of s cell (by size, we mean side length ). Case 1: and +1 are edge neighbors. In this case, the connectiity between and +1 in Q may come from either the application of a template (applytemplate) at some leel of recursion, or the application of the stitching step (stitchchains) at some leel of recursion. We consider different possibilities based on the ratio s i : s i+1, which may be 1 : 1, 1 : 2, or 2 : 1. Configurations for these cases are shown in Figs. 9, 10, and 11, respectiely. Each of these figures indicates the minimum internal angle in A i along α i. Note that each of them is well aboe We depict only distinct α i that differ in either the number of edges, or the angles at the ertices (that is, we do not show other, symmetric configurations that lead to the same α i ) (i) (ii) (iii) Fig. 9: Configurations for α i when s i : s i+1 is Fig. 10: Configs. for α i when s i : s i+1 is 1 : 1. (i) and (ii) come from stitching base 1 : 2. (i) and (ii) from applytemplate and stitching base cases. cases, and (iii) from stitching merge steps. (iii) from stitching merge steps. (i) (ii) (iii) Case 2: and +1 are corner neighbors. In this case, the connectiity between and +1 in Q may come from the application of applytemplate, the application of

10 10 F. Betul Atalay, Suneeta Ramaswami, and Dianna Xu (i) (iii) (i) (ii) Fig. 11: Configurations for α i when s i : s i+1 is 2 : 1. (i) and (ii) come from applytemplate and stitching base cases. (iii) and (i) occur only in the stitching base cases. stitchchains, or through a center quad. The center quad is the quadrilateral formed at the center, i.e. the meeting point of the four quadrants, after a general template (ref. Fig. 3) is applied during the recursie step. Since and +1 are corner neighbors, the ratio s i : s i+1 can be 1 : 1, 1 : 2, 2 : 1, 1 : 4, or 4 : 1. We consider the case of center quads first, and then consider templates and stitchings. Case 2.1: α i contains center quad edges. Let s be the size of the cell adjacent to as well as +1 and lying aboe +1. Possible configurations for α i when s i : s : s i+1 1 : 1 : 1 are shown in Fig. 12. In 12(i), 12(i), and 12(i), the point in the cell adjacent to and +1 may be either a cell center or an extra point of a larger cell. When s i : s : s i+1 1 : 2 : 1 or s i : s : s i+1 1 : 1 2 : 1, possible configurations of α i are shown in Figs. 13 and 14, respectiely (i) (ii) (iii) (i) () (i) Fig. 12: s i : s : s i+1 1 : 1 : Fig. 13: s i : s : s i+1 1 : 2 : 1 Fig. 14: s i : s : s i+1 1 : 1 2 : 1 When s i : s : s i+1 1 : 1 : 2, possible configurations of α i are shown in Fig. 15. For the case when s i : s : s i+1 2 : 1 : 1, the α i are obtained by reflections about the line y = x of those in Case Hence the minimum internal angle shown in Fig. 15 holds here as well. Similarly, Fig. 16 depicts α i when s i : s : s i+1 1 : 2 : 2 and the chains in this figure are reflections about the line y = x of possible α i when s i : s : s i+1 2 : 2 : 1 Finally, Fig. 17 shows possible configurations of α i when s i : s : s i+1 1 : 2 : 4. For the case when s i : s : s i+1 4 : 2 : 1, the α i are obtained by 180 rotations of those in Fig. 17.

11 Quadrilateral Meshes with Bounded Minimum Angle Fig. 15: s i : s : s i+1 1 : 1 : Fig. 16: s i : s : s i+1 1 : 2 : Fig. 17: s i : s : s i+1 1 : 2 : 4 Case 2.2: α i constructed by application of applytemplate or stitchchains. All new configurations of α i that occur by a template application, or a stitching step at some leel of recursion are listed. By new, we mean configurations that do not appear in Figs Note that when and +1 are connected ia templates or stitchings, s i : s i+1 is 1 : 1 (see Fig. 18), 1 : 2, (see Fig. 19) or 2 : 1 (reflections about the line of y = x of the α i in Fig. 19), but not 1 : 4 or 4 : c i (i) (ii) (iii) (i) () (i) Fig. 18: s i : s i+1 1 : (i) (ii) (iii) (i) () (i) (ii) Fig. 19: s i : s i+1 1 : 2 While Figs all depict and +1 in the southwest and northeast quadrants respectiely, note that each of the α i in these figures has a 90 rotational symmetry corresponding to and +1 in the northwest and southeast quadrants, which does not change the minimum internal angles indicated in those figures. It follows from the aboe case analysis that α i has at most four edges. We now describe how to quadrangulate each polygonal region A i = ( i, ) α i (+1, i+1 ) ( i+1, i ) independently for 1 i m. Before doing this, we first

12 12 F. Betul Atalay, Suneeta Ramaswami, and Dianna Xu show that rather than using the ertical projections i and i+1, we may instead use perpendicular projections of and +1 onto edge e (Lemma 6). This allows us to proe angle bounds for quadrangulating A i that are independent of the angle that e makes with the horizontal (recall that this is between 45 and 45 ). Lemma 5. Let δ i be the signed angle (in degrees) between +1 and the positie x-axis. Then abs(δ i ) {0, 18.43, 45, 71.57, 90}. Proof: Since and +1 are both cell centers in QT, and we know from Lemma 2 that they are edge or corner neighbors, it follows that there are a constant number of possibilities for δ i : If and +1 are edge neighbors with s i = s i+1, then δ i is either 0 or 90. If and +1 are edge neighbors with s i s i+1, then tan(δ i ) = 1 3, i.e., abs(δ i ) = 18.43, or tan(δ i ) = 3, i.e., abs(δ i ) = If and +1 are corner neighbors, then abs(δ i ) = 45. Let θ be the signed angle made by e with the positie x-axis. The alue of θ determines the range of possibilities for δ i. This is because of our definition of the quadtree chain, which specifies that either cell (resp. +1 ) has center aboe e and is intersected by e, or it is the north/west neighbor of a cell intersected by e whose center lies below e. Table 1 summarizes the possible alues of δ i for gien ranges of θ. This relationship also allows us to proe Lemma 6, stated here without proof. Range of θ Values of δ i θ < δ i 90 0 < θ < δ i < < θ δ i < θ δ i Table 1: Range of alues for θ and δ i. Lemma 6. Let p i be the perpendicular projection of on e, and i the ertical projection of on e. Assume e makes an angle between 45 and 45 with the positie x axis. Then for all 1 i < m, +1 lies outside the triangle (p i i ). We know from Lemma 3 that for all 1 i m, α i does not intersect i or +1 i+1. Furthermore, we know from the proof of Lemma 4 that α i lies aboe +1 (that is, it does not intersect the region bounded by i i+1 +1 ). Therefore, Lemma 6 implies that α i does not intersect p i or +1 p i+1 either. We redefine polygon A i to be (p i, ) α i (+1, p i+1 ) (p i+1, p i ) (that is, it is defined by the perpendicular projections rather than the ertical ones). Lemmas 7 and 8 establish bounds on some angles in A i when α i has one and two edges, respectiely. Lemma 7. Let φ 1 = p i +1 and κ 1 = +1 p i+1. Then min{φ 1, κ 1 } Proof: Refer to Fig. 20(a). Since φ 1 = 90 θ + δ i and κ 1 = 90 + θ δ i (recall θ and δ i are signed angles), and the fact (θ δ i ) (refer to Table 1), it follows that φ and κ

13 e (a) θ δ i φ 1 κ 1 +1 θ p i p i+1 (b) Quadrilateral Meshes with Bounded Minimum Angle 13 φ 1 φ 2 κ 1 κ 2 +1 p i p i+1 e (c) (d) (e) Fig. 20: (a) min{φ 1, κ 1} (b) min{φ 1 + φ 2, κ 1 + κ 2} (c) Shaded angle is (d)-(e) Shaded angle is Lemma 8. Suppose α i has two edges, and +1. Let φ 1 = p i +1, κ 1 = +1 p i+1, φ 2 = +1, and κ 2 = +1. Then (i) min{φ 1, κ 1 } > and (ii) min{φ 1 + φ 2, κ 1 + κ 2 } Proof: (i) From Lemma 7 we know that min{φ 1, κ 1 } To see that it must be strictly greater, note that if min{φ 1, κ 1 } = then abs(θ δ i ) = From Lemma 5 and Table 1, it can be seen that abs(θ δ i ) = when (a) θ = and δ i = 90, which is impossible because and +1 are directly connected wheneer δ i = 90, or (b) θ = 0 and δ i = 71.57, which is also impossible because θ must be strictly greater than 0 wheneer δ i = (ii) Refer to Fig. 20. First obsere that if min{φ 2, κ 2 } 18.43, then part (i) implies the claim. Hence assume that min{φ 2, κ 2 } < The only configurations of α i for which min{φ 2, κ 2 } < are shown in Fig. 20(c)-(e). We use the angle dependency in Table 1 to proe that these configurations imply min{φ 1 + φ 2, κ 1 + κ 2 } Further details are omitted here. Lemma 9. For 1 i m 1, the simple polygon A i = (p i, ) α i (+1, p i+1 ) (p i+1, p i ) can be quadrangulated with at most fie quadrilaterals with a minimum angle of Proof: Since α i has at most four edges (Lemma 4), we hae four cases. Case 1: α i has one edge. In this case, A i is already a quadrilateral. The fact that all angles of A i are at least follows from Lemma 7. Case 2: α i has two edges. Let and +1 be the two edges of α i. Let φ 1, κ 1, φ 2, and κ 2 be as in Fig. 20(b). Let γ = +1. Obsere that γ because the edges of α i come from Q. We quadrangulate A i according to the angles φ 2 and κ 2. (a) p i φ 2 φ 2 φ 2 s φ 2 κ 2 s κ 2 κ 2 p p s +1 i i+1 c p i i+1 p p e (b) p i+1 e (c) p p i+1 e (d) s p i+1 Fig. 21: α i has two edges. (a) min{φ 2, κ 2} (b) φ 2 < and κ (c) φ and κ 2 < (d) φ 2 < and κ 2 < κ 2 +1 p i+1 e

14 14 F. Betul Atalay, Suneeta Ramaswami, and Dianna Xu If min{φ 2, κ 2 } 18.43, place a Steiner point s on +1 and at the perpendicular projection p of s onto e. Connect s to, +1, and p to obtain a quadrangulation of A i. Perturb s towards p to obtain strictly conex quadrilaterals. We know from Lemma 8(i) that there is always a small perturbation of s that maintains all angles in the resulting quadrangulation at or aboe See Fig. 21(a). If min{φ 2, κ 2 } < 18.43, the placement of Steiner points depends on which of φ 2 and κ 2 are smaller than (assume wlog that φ 1 + φ 2 κ 1 + κ 2 ). If exactly one of φ 2 or κ 2 is less than 18.43, then Lemma 8 guarantees a quadrangulation of A i with the required angle bounds. We omit details and refer to Fig. 21(b)-(c). If both φ 2 and κ 2 are less than 18.43, the only possible configuration for α i is shown in Fig. 20(e). Obsere that in this case, can see e. Let s be the perpendicular projection of onto e, unless 0 < θ < 18.43, in which case let s be the ertical projection of onto e. Connect to s to obtain a quadrangulation of A i in which all angles satisfy the lower bound. Case 3: α i has three edges. The method used to quadrangulate A i depends on the number of reflex internal ertices of α i, which is zero or one (note that since α i lies aboe +1, it is not possible for both internal angles to be reflex): If the two internal angles along α i are both conex, draw an edge between and +1, which quadrangulates A i with two quadrilaterals. Some examples of such α i can be seen in Fig. 10(iii) and 19(ii). In all such cases, Lemma 7 guarantees that all angles in the quad below +1 is at least The quad aboe +1 has a minimum angle of See Fig. 22(a). If one of the internal angles along α i is reflex, and +1 must be corner neighbors. Let r be the reflex ertex. r either lies on the segment +1 (Fig. 22(b)), or belongs to the quadtree cell N( ) adjacent to and +1 and lying aboe +1 (Fig. 22(c)). Seeral examples of the former appear in Figs For the latter, see Fig. 18()-(i) and Fig. 19(i). We can show that in both cases, A i can be decomposed into a quadrilateral and a pentagon that is further decomposed into three quads with the required minimum angle bounds. Details are omitted here. c +1 i e e e (a) (b) r r (c) e (a) e (b) Fig. 22: α i has three edges. Fig. 23: α i has four edges. Case 4: α i has four edges. α i is classified according to the three internal ertices: If the three internal ertices consist of two reflex ertices separated by a conex ertex (e.g., Fig. 18(i)), the reflex ertices always lie on +1. Insert

15 Quadrilateral Meshes with Bounded Minimum Angle 15 edges from each reflex ertex to its perpendicular projection onto e. This decomposes A i into two quads and a pentagon. Lemmas 7 and 8 proide the required minimum angle bounds. See Fig. 23(a). If the three internal ertices consist of two conex ertices separated by a reflex ertex (Fig. 18(ii)), decompose A i into four quads as shown in Fig. 23(b). Note that this decomposition has the required minimum angle bounds regardless of the alue of θ. This completes the proof that A i can be quadrangulated with at most fie quadrilaterals with a minimum angle of Edge separation conditions for quadtree Eery edge e of the polygon P defines a chain of edges gien by 1 i m α i. From this chain, we obtain the polygons A i, each of which is then quadrangulated as described aboe. In order to conduct this process independently for eery edge of the polygon, we impose an edge separation condition on QT. The edge separation condition requires that all quadrangulation chains 1 i m α i defined by the edges of the polygon be disjoint from each other. Recall that these chains do not start in the cell containing the segment endpoint, but rather in one adjacent to it. This allows quadrangulation chains to be separated completely, except in the 5 5 grid of cells around each polygon ertex. In the worst case, the edge separation condition requires that eery cell intersected by a polygon edge be surrounded by a 3 3 grid of empty cells, but in practice, this requirement does not apply uniformly across the entire segment. Connecting quadtree chains around polygon ertices For eery edge of the polygon P, the quadtree chain starts and ends at a cell center within the 3 3 grid of quadtree cells that is guaranteed to exist around each of its endpoints. Let be a ertex of P and let e and f be the two oriented edges incident on (the interior of P lies to their left). Let u be the last quadtree chain ertex for edge e and let w be the first quadtree chain ertex for edge f. Note that u and w are both cell centers in the 3 3 grid around. Let ū and w be the perpendicular projections of u and w onto e and f, respectiely. Let E be a sequence of edges connecting u to w in the 3 3 grid. The region around ertex is meshed by quadrangulating the polygon P defined by the edges ū, ūu, E, w w, w. The method used to quadrangulate P depends on the number of edges in E, which is between one and seen (inclusie). Refer to Fig. 24 for an illustration of some cases. The underlying 3 3 grid is used to proe the following lemma, stated here without proof. Lemma 10. P can be decomposed into at most seen quadrilaterals with a minimum angle of Figure 25 shows quadrangulation chains 1 i m α i for some edges of a polygon (the entire polygon is shown in Fig. 28). Quadtree chain ertices are highlighted. Summary of algorithm. We summarize in Figure 26 the algorithm to quadrangulate the interior of a non-acute simple polygon P of n edges e 1, e 2,..., e n and

16 16 F. Betul Atalay, Suneeta Ramaswami, and Dianna Xu e u w f (a) u e w f u e (b) f w u e x w f (c) u x u e f (d) w Fig. 24: Connecting at corners. Number of edges in E is (a) one, (b) two, (c)-(d) three or more. Quadrangulate(P, n): QT Quadtree decomposition satisfying edge separation conditions for ertices of P Q Quadrangulation resulting from applytemplate on QT. for e i {e 1, e 2,..., e n} α(e i) Quadrangulation chain for e i Q(e i) Quadrangulation of region bounded by e i and α(e i), as gien by Lemma 9. Q( i) Quadrangulation of corner polygon P i, as gien by Lemma 10. Q S 1 i n Q(ei) S 1 i n Q(i) return Q ( Q (P Q )) Fig. 25: Quadrangulation chains. Fig. 26: Summary of algorithm. ertices 0, 1,..., n 1, where e i = ( i 1, i ) (where n = 0 ). The resulting quadrilaterals hae a minimum angle bound of Theorem 2. Gien a quadtree decomposition with N quadtree cells satisfying the edge separation condition for a simple polygon P, Quadrangulate(P, n) constructs a mesh for P with at most 5N quadrilaterals in which eery angle is at least General Simple Polygons Let P be a general simple polygon containing acute angles. We first conert P into a polygon that contains only obtuse angles by cutting off the acute angle ertices. Let a be an acute angle ertex of P. Let θ, 0 θ < 90, be the angle at that ertex. Let be a point on the angle bisector of a, and let p and q be the perpendicular projections of onto the two edges incident at a θ p p 2 p 3 32 p p 1 γ 1 2 γ q q Fig. 27: Handling acute angles in P. a. is chosen so that the quadrangular region apq does not contain any other ertices of the polygon P. Cut all such regions apq from P. Let B be the polygon resulting from this procedure. Construct a quadrilateral mesh for B using the algorithm in Section 3.1. Obsere that now there might be Steiner points on p (e.g., 1, 2, 3 in Fig. 27) and q. These are used to quadrangulate the region apq with the required angle bounds. We omit details and refer to Fig. 27 for an illustration of the quadrangulation.

17 4 Conclusion Quadrilateral Meshes with Bounded Minimum Angle 17 Sample meshes generated by our algorithm are shown in Figures 28 and 29. Obsere that in these examples the ratio of the number of quadrilaterals to the number of quadtree cells is less than one. The image on the right shows a zoomed-in portion of the mesh on the left. Fig. 28: Number of polygon edges: 19. Minimum mesh angle: Number of quadtree cells: Number of mesh ertices: 989. Number of mesh faces (quads): 841 Fig. 29: Number of polygon edges: 33. Minimum mesh angle: Number of quadtree cells: Number of mesh ertices: Number of mesh faces (quads): 1859 This paper presents the first known result on the generation of a quadrilateral mesh for the interior of a simple polygon (possibly with holes) in which eery new

18 18 F. Betul Atalay, Suneeta Ramaswami, and Dianna Xu angle in the mesh is bounded from below. The main open question resulting from this work is its extension to polygon interior as well as exterior. While our algorithm itself is applicable to the interior or the exterior of the polygon, the difficulty of adapting it to both lies in resoling mesh compatibility at the boundary without propagating the changes throughout the mesh. We are currently inestigating alternatie strategies to mesh the region bounded by quadtree chains on both sides of each polygon edge. Acknowledgments The authors would like to thank anonymous reiewers for helpful comments that sered to improe the presentation in the paper. References 1. D. J. Allman. A quadrilateral finite element including ertex rotations for plane elasticity analysis. Int. J. Numer. Meth. in Engineering, 26: , S. Benzley, E. Perry, K. Merkley, B. Clark, and G. Sjaardema. A Comparison of All Hexahedral and All Tetrahedral Finite Element Meshes for Elastic and Elastic-Plastic Analysis. In Proc. of the 4th International Meshing Roundtable, pages , M. Bern and D. Eppstein. Quadrilateral meshing by circle packing. In 6th IMR, pages 7 19, M. Bern, D. Eppstein, and J. Gilbert. Proably good mesh generation. J. Comp. Sys. Sci., 48: , T. Blacker and M. Stephenson. Paing: A new approach to automated quadrilateral mesh generation. Int. J. Numer. Meth. in Engineering, 32(4): , J. R. Brauer, editor. What eery engineer should know about finite element analysis. Marcel-Dekker, 2nd edition, S-W Cheng, T. Dey, E. Ramos, and T. Ray. Quality meshing for polyhedra with small angles. In Proc. 20th Annual Symposium on Computational Geometry, L.P. Chew. Guaranteed-quality mesh generation for cured surfaces. In Proc. of the 9th ACM Symposium on Computational Geometry, pages , L.P. Chew. Guaranteed-quality delaunay meshing in 3d. In Proc. of the 13th ACM Symposium on Computational Geometry, pages , B. P. Johnston, J. M. Sullian, and A. Kwasnik. Automatic conersion of triangular finite meshes to quadrilateral elements. Int. J. Numer. Meth. in Engineering, 31(1):67 84, G. L. Miller, D. Talmor, S-H. Teng, and N. Walkington. A delaunay based numerical method for three dimensions: Generation, formulation, and partition. In Proc. of the 27th ACM Symposium on the Theory of Computing, pages , S. Mitchell and S. Vaasis. Quality Mesh generation in Three Dimensions. In Proceedings of the 8th Annual Symposium on Computational Geometry, pages , S. Ramaswami, M. Siqueira, T. Sundaram, J. Gallier, and J. Gee. Constrained quadrilateral meshes of bounded size. Int. J. Comp. Geom. and Appl., 15(1):55 98, J. Shewchuk. Constrained Delaunay Tetrahedralizations and Proably Good Boundary Recoery. In Proc. of the 11th International Meshing Roundtable, pages , J. R. Shewchuk. Triangle: Engineering a 2D quality mesh generator and Delaunay triangulator. In Applied Comp. Geom.: Towards Geometric Engineering, olume LNCS 1148, O. C. Zienkiewicz and R. L. Taylor. The finite element method. McGraw-Hill, 1989.